arXiv subjects

Sourin Das

Publications and source records attributed to Sourin Das.

At least 37 records · Page 2Linked to original sources

arXiv2024

Fractional Conformal Map, Qubit Dynamics and the Leggett-Garg Inequality

Any pure state of a qubit can be geometrically represented as a point on the extended complex plane through stereographic projection. By employing successive conformal maps on the extended complex plane, we can generate an effective discrete-time evolution of the pure states of the qubit. This work focuses on a subset of analytic maps known as fractional linear conformal maps. We show that these maps serve as a unifying framework for a diverse range of quantum-inspired conceivable dynamics, including (i) unitary dynamics,(ii) non-unitary but linear dynamics and (iii) non-unitary and non-linear dynamics where linearity (non-linearity) refers to the action of the discrete time evolution operator on the Hilbert space. We provide a characterization of these maps in terms of Leggett-Garg Inequality complemented with No-signaling in Time (NSIT) and Arrow of Time (AoT) conditions.

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arXiv2023

Can Majorana zero modes in quantum Hall edges survive edge reconstruction?

Parafermion zero modes can be trapped in the domain walls of quantum Hall edges proximitized by superconductors and ferromagnets. The $ν= 1/3$ fractional quantum Hall side strip arising due to edge reconstruction of a $ν= 1$ edge doubles the number of topological sectors such that each of them is $Z_{2} \times Z_{2}$ degenerate. The many-body spectrum displays a $4π$ Josephson periodicity, with the states in each $Z_{2}$ being energetically decoupled. Signatures of the new states appear in the fractional Josephson current when the edge velocities are taken to be different.

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arXiv2023

Quantum spin Hall insulator in proximity with a superconductor: Transition to the Fulde-Ferrell-Larkin-Ovchinnikov state driven by a Zeeman field

We investigate the effects of introducing a boost (a Zeeman field parallel to the spin quantization axis) at the proximitized helical edge of a two-dimensional (2D) quantum spin Hall insulator. Our self-consistent analysis finds that a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting phase may emerge at the edge when the boost is larger than a critical value tied to the induced pairing gap. A non-trivial consequence of retaining the 2D bulk in the model is that this boundary FFLO state supports a finite magnetization as well as finite current (flowing along the edge). This has implications for a proper treatment of the ultra-violet cutoff in analyses employing the effective one-dimensional (1D) helical edge model. Our results may be contrasted with previous studies of such 1D models, which found that the FFLO phase either does not appear for any value of the boost (in non-self-consistent calculations), or that it self-consistently appears even for infinitesimal boost, but carries no current and magnetization.

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arXiv2023

Anomalous topology and synthetic flat band in multi-terminal Josephson Junctions

Andreev bound states trapped in a multi-terminal Josephson junction (JJ) can be assigned a synthetic band topology owing to their periodic dependence on the Josephson phase bias. We demonstrate that the BdG symmetry adds a twist to this topological character, i.e., gap closing points may or \textit{may not} correspond to change of Chern number, hence extending the standard paradigm for topological bands. We further show that the topology of Andreev bands depends only on the scattering matrix of the junction and is independent of the topological nature of superconductors forming the JJ hence indicating a universal behaviour of multi-terminal JJ. We also show that the chiral junction, supported by quantum Hall state at the junction region, leads to flat Andreev bands (implying absence of DC Josephson effects) that are devoid of Berry curvature ( implying absence of AC Josephson effects). Such electrically inert JJ may be useful for storage of quantum information in future quantum devices.

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arXiv2023

Multipartite entanglement and quantum error identification in $D$-dimensional cluster states

An entangled state is said to be $m$-uniform if the reduced density matrix of any $m$ qubits is maximally mixed. This is intimately linked to pure quantum error correction codes (QECCs), which allow not only to correct errors, but also to identify their precise nature and location. Here, we show how to create $m$-uniform states using local gates or interactions and elucidate several QECC applications. We first show that $D$-dimensional cluster states are $m$-uniform with $m=2D$. This zero-correlation length cluster state does not have finite size corrections to its $m=2D$ uniformity, which is exact both for infinite and for large enough but finite lattices. Yet at some finite value of the lattice extension in each of the $D$ dimensions, which we bound, the uniformity is degraded due to finite support operators which wind around the system. We also outline how to achieve larger $m$ values using quasi-$D$ dimensional cluster states. This opens the possibility to use cluster states to benchmark errors on quantum computers. We demonstrate this ability on a superconducting quantum computer, focusing on the 1D cluster state which, we show, allows to detect and identify 1-qubit errors, distinguishing $X$, $Y$ and $Z$ errors.

quant-ph↗

arXiv2023

Observing super-quantum correlations across the exceptional point in a single, two-level trapped ion

Quantum theory provides rules governing much of the microscopic world, and among its counter-intuitive consequences are correlations that exceed the bounds from local, classical theories. In two-level quantum systems - qubits - unitary dynamics theoretically limit these spatiotemporal quantum correlations, called Bell/Clauser-Horn-Shimony-Holt or Leggett-Garg inequalities, to $2\sqrt{2}$ or 1.5 respectively. Experiments with state-of-the-art qubits have approached the spatial, Bell and temporal, Leggett-Garg quantum correlation bounds. Here, using a dissipative, trapped $^{40}$Ca$^+$ ion governed by a two-level, non-Hermitian Hamiltonian, we observe temporal correlation values up to 1.703(4) for the Leggett-Garg parameter $K_3$, clearly exceeding the hitherto inviolable Lüder's bound of 1.5. These excesses occur across the exceptional point of the parity-time symmetric Hamiltonian responsible for the qubit's non-unitary, coherent dynamics. Distinct evolution speeds for antipodal qubit states, which violate the unified (Mendelstam-Tamm or Margolus-Levitin) bound $τ_{\textrm{QSL}}$ for the transit time based on quantum speed limit, result in the super-quantum $K_3$ values observed over a wide parameter range. Our results demonstrate that post-selected, coherent dynamics of non-Hermitian Hamiltonians pave the way for enhanced quantum correlations that exceed protocols based on unitary or dissipative dynamics.

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arXiv2023

Topology of quadrupolar Berry phase of a Qutrit

We examine Berry phase pertaining to purely quadrupolar state ($\langle ψ| \vec{S} | ψ\rangle = 0$) of a spin-$1$ system. Using the Majorana stellar representation of these states, we provide a visualization for the topological (zero or $π$) nature of such quadrupolar Berry phase. We demonstrates that the $π$ Berry phase of quadrupolar state is induced by the Majorana stars collectively tracing out a closed path (a great circle) by exchanging their respective positions on the Bloch sphere. We also analyse the problem from the perspective of dynamics where a state from the quadrupolar subspace is subjected to a static magnetic field. We show that time evolution generated by such Hamiltonian restricts the states to the quadrupolar subspace itself thereby producing a geometric phase (of the Aharonov-Anandan type) quantized to zero or $π$. A global unitary transformation which maps the quadrupolar subspace to the subspace of purely real states proves a natural way of understanding the topological character of this subspace and its connection to the anti-unitary symmetries.

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arXiv2023

Enhancement in tunneling density of states in Luttinger liquid -- role of non-local interaction

Power law suppression of local electronic tunneling density of states (TDOS) in the zero-energy limit is a hallmark of Luttinger liquid (LL) phase of the interacting 1-D electron system. We present a theoretical model which hosts LL state with the surprising feature of enhancement rather than suppression in local TDOS originating from non-local and repulsive density-density interactions. Importantly, we find enhancement of TDOS in the manifold of parameter space where the system is stable in the renormalization group (RG) sense. We argue that enhancement of TDOS along with RG stability is possible only when the system has broken parity symmetry about the position of local TDOS enhancement. Such a model could be realized on the edge states of a bi-layer quantum Hall system where both intra-layer and inter-layer density-density interactions are present mimicking the role of local and non-local interactions respectively.

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arXiv2023

Tunneling density of states in Luttinger Liquid in proximity to a superconductor: Effect of non-local interaction

A recent study have shown that it is possible to have enhancement, in contrast to an expected suppression, in tunneling density of states (TDOS) in a Luttinger liquid (LL) which is solely driven by the non-local density-density interactions. Also, it is well known that a LL in proximity to a superconductor (SC) shows enhancement in TDOS in the vicinity of the junction in the zero energy limit. In this paper, we study the interplay of nonlocal density-density interaction and superconducting correlations in the TDOS in the vicinity of the SC-LL junction, where the LL maybe realized on the edge of an integer or a fractional quantum Hall state. We show that the TDOS enhancement in a LL quantum wire (QW) due to superconducting correlation at the junction can be farther amplified in presence of appropriate choice of non-local interaction parameters, which is demonstrated through a neat algebraic expression in the weak interaction limit. We also show that, in the full parameter regime comprising both the local and the non-local interaction, the region of enhanced TDOS for LL junction with "superconducting" boundary condition and that of "non-superconducting charge conserving" boundary condition (discussed in Phys. Rev. B 104, 045402 (2021)) are mutually exclusive. We show that this fact can be understood in terms of symmetry relation established between the superconducting and non-superconducting sectors of the theory. We compare the dependence of the proximity induced pair potential and TDOS as a function of distance $x$ from the junction. We demonstrate that the dependence of the spatial power law exponent for the `TDOS(x)/TDOS(x -> 0)' and the `pair-potential(x)/pair-potential(x -> 0)' are distinct function of the various local and non-local interaction parameters, which implies that the TDOS enhancement can not be directly attributed to the proximity induced pair potential in the LL.

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arXiv2022

Volkov-Pankratov states on the edge of a quantum spin Hall system

Volkov-Pankratov (VP) states are a family of sub-gap states which appear at the smooth interface/domain wall between topologically distinct gapped states. We study the emergence of such states in the edge spectrum of a quantum spin Hall system subjected to a smoothly varying mass term (Zeeman field) that switches sign at a given spatial point. Both the VP states at non-zero energy and the zero energy Jackiw-Rebbi mode stay localized at the interfacial region, however, the former feature several distinctive signatures compared to the latter such as non-trivial spin textures that can be characterized by a winding number in real space. On applying an electric field, the texture deforms leaving its winding number unaltered. Moreover, the VP states exhibit an intriguing interplay between the electric and the magnetic field with a collapse of the spectrum onto the zero mode when they are equal in magnitude. Quantum transport simulations on a 2D lattice model are performed to undergird our theoretical prediction.

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arXiv2022

Spontaneous fractional Josephson current from parafermions

We study a parafermion Josephson junction (JJ) comprising a pair of counter-propagating edge modes of two quantum Hall (QH) systems, proximitized by an s-wave superconductor. We show that the difference between the lengths (which can be controlled by external gates) of the two counter-propagating chiral edges at the Josephson junction, can act as a source of spontaneous phase bias. For the Laughlin filling fractions, $ν= 1/m,~ m \in 2\mathbb{Z}+1$, this leads to an electrical control of either Majorana $(m=1)$ or parafermion $(m\neq 1)$ zero modes.

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arXiv2022

$n$-qubit states with maximum entanglement across all bipartitions: A graph state approach

We discuss the construction of $n$-qubit pure states with maximum bipartite entanglement across all possible choices of $k$ vs $n-k$ bi-partitioning, which implies that the Von Neumann entropy of every $k$-qubit reduced density matrix corresponding to this state should be $k \ln 2 $. Such states have been referred to as $k$-uniform, $k$-MM states. We show that a subset of the 'graph states' satisfy this condition, hence providing a recipe for constructing $k$-uniform states. Finding recipes for construction of $k$-uniform states using graph states is useful since every graph state can be constructed starting from a product state using only controlled-$Z$ gates. Though, a priori it is not clear how to construct a graph which corresponds to an arbitrary $k$-uniform state, but in particular, we show that graphs with no isolated vertices are $1$-uniform. Graphs organized as a circular linear chain corresponds to the case of $2$-uniform state, where we show that the minimum number of qubits required to host such a state is $n=5$. $3$-uniform states can be constructed by forming bi-layer graphs with $n/2$ qubits ($n=2\mathbb{Z}$) in each layer, such that each layer forms a fully connected graph while inter-layer connections are such that the vertices in one layer has a one to one connectivity to the other layer. $4$-uniform states can be formed by taking 2D lattice graphs( also referred elsewhere as a 2D cluster Ising state ) with periodic boundary conditions along both dimensions and both dimensions having at least $5$ vertices.

quant-ph↗

arXiv2022

Non-local spin entanglement in a fermionic chain

An effective two-spin density matrix (TSDM) for a pair of spin-$1/2$ degree of freedom, residing at a distance of $R$ in a spinful Fermi sea, can be obtained from the two-electron density matrix following the framework prescribed in Phys. Rev. A 69, 054305 (2004). We note that the single spin density matrix (SSDM) obtained from this TSDM for generic spin-degenerate systems of free fermions is always pinned to the maximally mixed state $i.e.$ $(1/2) \ \mathbb{I}$, independent of the distance $R$ while the TSDM confirms to the form for the set of maximally entangled mixed state (the so called "X-state") at finite $R$. The X-state reduces to a pure state (a singlet) in the $R\rightarrow 0$ limit while it saturates to an X-state with largest allowed value of von-Neumann entropy of $2 \ln2$ as $R \rightarrow \infty$ independent of the value of chemical potential. However, once an external magnetic field is applied to lift the spin-degeneracy, we find that the von-Neumann entropy of SSDM becomes a function of the distance $R$ between the two spins. We also show that the von-Neumann entropy of TSDM in the $R\rightarrow \infty$ limit becomes a function of the chemical potential and it saturate to $2 \ln2$ only when the band in completely filled unlike the spin-degenerate case. Finally we extend our study to include spin-orbit coupling and show that it does effect these asymptotic results. Our findings are in sharp contrast with previous works which were based on continuum models owing to physics which stem from the lattice model.

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arXiv2022

Chiral detection of Majorana bound states at the edge of a quantum spin Hall insulator

A hybrid setup consisting of a superconductivity-proximitized quantum spin Hall (QSH) insulator and a quantum anomalous Hall (QAH) insulator is proposed for chiral injection of electrons into the Majorana bound state (MBS). An unexplored region of the phase space involving the exchange field induced boost of the helical edge state is then proposed for the detection of the MBS. 2-D transport simulations of our proposed setup is compared with the corresponding setup in the absence of the QAH region, when moderate disorder and a small but finite bulk out-of-plane magnetic field and a Rashba field are included. The remarkable contrast between the two results demonstrates the possibility for an unprecedented immunity from disorder-induced masking of the MBS detection in our proposed setup.

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arXiv2022

Essential role of quantum speed limit in violation of Leggett-Garg inequality across a PT-transition

We study Leggett-Garg inequality (LGI) of a two level system (TLS) undergoing non-Hermitian dynamics governed by a non-linear Bloch equation (derived in J. Phys. A: Math. Theor. 54, 115301 (2021)) across a PT-transition. We present an algebraic identification of the parameter space for the maximum violation of LGI (in particular $K_{3}$). In the PT-symmetric regime the maximum allowed value for $K_{3}$ is always found to be greater than the quantum bound (Lüders bound) of $3/2$ but it does not reach the algebraic maximum of $K_{3}=3$ in general. However, in the limit where PT-symmetry breaking parameter approaches the exceptional point from the PT-symmetric side, $K_{3}$ is found to asymptotically approach its algebraic maximum of 3. In contrast, the maximum value of $K_{3}$ always reaches its algebraic maximum in the PT-broken phase $i.e.$ $K_{3}\rightarrow 3$. We find that (i) the speed of evolution (SOE) must reach its maximum value (in the parameter space of initial state and the time interval between successive measurements) to facilitate the value of $K_{3} \rightarrow 3$, (ii) together with the constraint that its minimum value must run into SOE equals to zero during the evolution of the state. In fact we show that the minimum speed of evolution can serve as an order parameter which is finite on the PT-symmetric side and identically zero on the PT-broken side. Finally, we discuss a possible experimental realization of this dynamics by quantum measurement followed by post-selection procedure in a three level atom coupled to cavity mode undergoing a Lindbladian dynamics.

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arXiv2021

Thermoelectric response of Josephson junction: from ballistic to disordered

It is known that Josephson junction (JJ) hosting scattering centers with energy-dependent scattering amplitudes which breaks the $ω\rightarrow-ω$ symmetry (where $ω$ is the excitation energy of electron about the Fermi level) exhibits finite thermoelectric response. In contrast, here we show that even in a ballistic JJ this symmetry is broken and it leads to a non-zero thermoelectric response when the junction length is of the order of coherence length of the superconductor and the corresponding thermoelectric coefficient confirms to the universal sinusoidal dependence on $ϕ_{12}$, where $ϕ_{12}$ is the superconducting phase bias. In presence of multiple scatterers in the junction region, we have numerically shown that the sign of the even-in-$ϕ_{12}$ part of the thermoelectric coefficient fluctuates violently from one disorder configuration to another hence averaging to vanishingly small values while the odd part tends towards the universal sinusoidal dependence on $ϕ_{12}$ as we approach the large disorder limit under disorder averaging.

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arXiv2021

Leggett-Garg inequality in Markovian quantum dynamics: role of temporal sequencing of coupling to bath

We study Leggett-Garg inequalities (LGIs) for a two level system (TLS) undergoing Markovian dynamics described by unital maps. We find analytic expression of LG parameter $K_{3}$ (simplest variant of LGIs) in terms of the parameters of two distinct unital maps representing time evolution for intervals: $t_{1}$ to $t_{2}$ and $t_{2}$ to $t_{3}$. We show that the maximum violation of LGI for these maps can never exceed well known Lüders bound of $K_{3}^{L\ddot{u}ders}=3/2$ over the full parameter space. We further show that if the map for the time interval $t_{1}$ to $t_{2}$ is non-unitary unital then irrespective of the choice of the map for interval $t_{2}$ to $t_{3}$ we can never reach Lüders bound. On the other hand, if the measurement operator eigenstates remain pure upon evolution from $t_{1}$ to $t_{2}$, then depending on the degree of decoherence induced by the unital map for the interval $t_{2}$ to $t_{3}$ we may or may not obtain Lüders bound. Specifically, we find that if the unital map for interval $t_{2}$ to $t_{3}$ leads to the shrinking of the Bloch vector beyond half of its unit length, then achieving the bound $K_{3}^{L\ddot{u}ders}$ is not possible. Hence our findings not only establish a threshold for decoherence which will allow for $K_{3} = K_{3}^{L\ddot{u}ders}$, but also demonstrate the importance of temporal sequencing of the exposure of a TLS to Markovian baths in obtaining Lüders bound.

quant-ph↗

arXiv2021

Temperature dependent equilibration of spin orthogonal quantum Hall edge modes

Conductance of the edge modes as well as conductance across the co-propagating edge modes around the ν= 4/3, 5/3 and 2 quantum Hall states are measured by individually exciting the modes. Temperature dependent equilibration rates of the outer unity conductance edge mode are presented for different filling fractions. We find that the equilibration rate of the outer unity conductance mode at ν= 2 is higher and more temperature sensitive compared to the mode at fractional filling 5/3 and 4/3. At lowest temperature, equilibration length of the outer unity conductance mode tends to saturate with lowering filling fraction νby increasing magnetic field B. We speculate this saturating nature of equilibration length is arising from an interplay of Coulomb correlation and spin orthogonality.

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