arXiv subjects
Publications and source records attributed to Tanmoy Bera.
Let $\mathcal{P}$ be a subset of primes and for each prime $p\in \mathcal{P}$, consider a subset $\mathcal{L}_p$ of $\mathbb{Z}/p\mathbb{Z}$. We provide restriction estimates with integers $\leq N$ sifted by $(\mathcal{L}_p)_{\substack{p\leq z\\ p\in \mathcal{P}}}$. This generalizes a result of Green-Tao [3] on the restriction estimates.
In this article we prove that for a strictly increasing sequence $(a_n)$ of natural numbers, if the additive energy of $\{a_n:n\leq N\}$ is less than $N^3/(\log N)^C$ for some $C\geq14.71,$ then $(\{a_n\alpha\})$ has Poissonian pair correlation for almost all $\alpha\in\mathbb{R}.$ This provides a lower bound for the exponent $C$ in the additive energy bound established by Bloom and Walker [4].
In this article, we examine the Poissonian pair correlation (PPC) statistic for higher-dimensional real sequences. Specifically, we demonstrate that for $d\geq 3$, almost all $(α_1,\ldots,α_d) \in \mathbb{R}^d$, the sequence $\big(\{x_nα_1\},\dots,\{x_nα_d\}\big)$ in $[0,1)^d$ has PPC conditionally on the additive energy bound of $(x_n).$ This bound is more relaxed compared to the additive energy bound for one dimension as discussed in [1]. More generally, we derive the PPC for $\big(\{x_n^{(1)}α_1\},\dots,\{x_n^{(d)}α_d\}\big) \in [0,1)^d$ for almost all $(α_1,\ldots,α_d) \in \mathbb{R}^d.$ As a consequence we establish the metric PPC for $(n^{θ_1},\ldots,n^{θ_d})$ provided that all of the $θ_i$'s are greater than two.
Cavity-electromechanical systems are extensively used for sensing and controlling the vibrations of mechanical resonators down to their quantum limit. The nonlinear radiation-pressure interaction in these systems could result in an unstable response of the mechanical resonator showing features such as frequency-combs, period-doubling bifurcations and chaos. However, due to weak light-matter interaction, typically these effects appear at very high driving strengths. By using polariton modes formed by a strongly coupled flux-tunable transmon and a microwave cavity, here we demonstrate an electromechanical device and achieve a single-photon coupling rate $g_0/2π$ of $160~$kHz, which is nearly 4\% of the mechanical frequency $ω_m$. Due to large $g_0/ω_m$ ratio, the device shows an unstable mechanical response resulting in frequency combs in sub-single photon limit. We systematically investigate the boundary of the unstable response and identify two important regimes governed by the optomechanical backaction and the nonlinearity of the electromagnetic mode. Such an improvement in the single-photon coupling rate and the observations of microwave frequency combs at single-photon levels may have applications in the quantum control of the motional states and critical parametric sensing. Our experiments strongly suggest the requirement of newer approaches to understand instabilities.
In this note, we extend the notion of minimal gaps to the higher dimensional sequences. We bound the minimal gap for $(\{\boldsymbol{a}_n\boldsymbol{\alpha}\}),$ $(\{a_n\boldsymbol{\alpha}\})$ and $(\{\boldsymbol{a}_n\cdot\boldsymbol{\alpha}\})$ in terms of cardinality of the difference set of $a_n$ and $\boldsymbol{a}_n,$ where $a_n$ and $a_n^{(1)},\dots,a_n^{(d)}$ are sequences of distinct integers.
With artificially engineered systems, it is now possible to realize the coherent interaction rate, which can become comparable to the mode frequencies, a regime known as ultrastrong coupling (USC). We experimentally realize a cavity-electromechanical device using a superconducting waveguide cavity and a mechanical resonator. In the presence of a strong pump, the mechanical-polaritons splitting can nearly reach 81% of the mechanical frequency, overwhelming all the dissipation rates. Approaching the USC limit, the steady-state response becomes unstable. We systematically measure the boundary of the unstable response while varying the pump parameters. The unstable dynamics display rich phases, such as self-induced oscillations, period-doubling bifurcation, period-tripling oscillations, and ultimately leading to the chaotic behavior. The experimental results and their theoretical modeling suggest the importance of residual nonlinear interaction terms in the weak-dissipative regime.
In this article we study the pair correlation statistic for higher dimensional sequences. We show that for any $d\geq 2$, strictly increasing sequences $(a_n^{(1)}),\ldots, (a_n^{(d)})$ of natural numbers have metric Poissonian pair correlation with respect to sup-norm if their joint additive energy is $O(N^{3-δ})$ for any $δ>0$. Further, in two dimension, we establish an analogous result with respect to $2$-norm. As a consequence, it follows that $(\{nα\}, \{n^2β\})$ and $(\{nα\}, \{[n\log^An]β\})$ ($A \in [1,2]$) have Poissonian pair correlation for almost all $(α,β)\in \mathbb{R}^2$ with respect to sup-norm and $2$-norm. This gives a negative answer to the question raised by Hofer and Kaltenböck [15]. The proof uses estimates for 'Generalized' GCD-sums.
Hybrid devices based on the superconducting qubits have emerged as a promising platform for controlling the quantum states of macroscopic resonators. The nonlinearity added by a qubit can be a valuable resource for such control. Here we study a hybrid system consisting of a mechanical resonator longitudinally coupled to a transmon qubit. The qubit readout can be done by coupling to a readout mode like in c-QED setup. The coupling between the mechanical resonator and transmon qubit can be implemented by modulation of the SQUID inductance. In such a tri-partite system, we analyze the steady-state occupation of the mechanical mode when all three modes are dispersively coupled. We use the quantum-noise and the Lindblad formalism to show that the sideband cooling of the mechanical mode to its ground state is achievable. We further experimentally demonstrate that measurements of the thermomechanical motion is possible in the dispersive limit, while maintaining a large coupling between qubit and mechanical mode. Our theoretical calculations suggest that single-photon strong coupling is within the experimental reach in such hybrid devices.
Superconducting qubits utilize the strong non-linearity of the Josephson junctions. Control over the Josephson nonlinearity, either by a current bias or by the magnetic flux, can be a valuable resource that brings tunability in the hybrid system consisting of superconducting qubits. To enable such a control, here we incorporate a fast-flux line for a frequency tunable transmon qubit in 3D cavity architecture. We investigate the flux-dependent dynamic range, relaxation from unconfined states, and the bandwidth of the flux-line. Using time-domain measurements, we probe transmon's relaxation from higher energy levels after populating the cavity with $\approx 2.1\times10^4$ photons. For the device used in the experiment, we find a resurgence time corresponding to the recovery of coherence to be 4.8~$μ$s. We use a fast-flux line to tune the qubit frequency and demonstrate the swap of a single excitation between cavity and qubit mode. By measuring the deviation in the transferred population from the theoretical prediction, we estimate the bandwidth of the flux line to be $\approx$~100~MHz, limited by the parasitic effect in the design. These results suggest that the approach taken here to implement a fast-flux line in a 3D cavity could be helpful for the hybrid devices based on the superconducting qubit.
Control over the quantum states of a massive oscillator is important for several technological applications and to test the fundamental limits of quantum mechanics. Addition of an internal degree of freedom to the oscillator could be a valuable resource for such control. Recently, hybrid electromechanical systems using superconducting qubits, based on electric-charge mediated coupling, have been quite successful. Here, we realize a hybrid device, consisting of a superconducting transmon qubit and a mechanical resonator coupled using the magnetic-flux. The coupling stems from the quantum-interference of the superconducting phase across the tunnel junctions. We demonstrate a vacuum electromechanical coupling rate up to 4 kHz by making the transmon qubit resonant with the readout cavity. Consequently, thermal-motion of the mechanical resonator is detected by driving the hybridized-mode with mean-occupancy well below one photon. By tuning qubit away from the cavity, electromechanical coupling can be enhanced to 40 kHz. In this limit, a small coherent drive on the mechanical resonator results in the splitting of qubit spectrum, and we observe interference signature arising from the Landau-Zener-Stückelberg effect. With improvements in qubit coherence, this system offers a novel platform to realize rich interactions and could potentially provide full control over the quantum motional states.