Switching the DC Response of an AC-driven Quantum Many-body system
Arnab Das Indian Association for the Cultivation of Science, Kolkata
Periodically Driven Transverse Ising Chain The Hamiltonian:
Dynamics: Schrodinger Evolution (T = 0) The Response: Transverse Magnetization 𝑚𝑧 Any net DC magnetization to the Purely AC Driving in the t → ∞ limit in the Non-Adiabatic case?
Classical “Dynamical Hysteresis” Response can “freeze” if the driving is too fast. Faster you drive, more frozen is the response dynamics Q increases monotonically with 𝝎
B. K. Chakrabarti and M. Acharyya, Rev. Mod. Phys. 71 847 (1999).
Dynamical Many-Body Freezing (DMF) Parameter Regime: h0, ω ≫ J (Fast and Strong Driving) Initial Condition: Ground State at 𝑡 = 0; 𝑚𝑧 ~ 1 Q = 1 => Complete Freezing of 𝑚𝑧 .
Q = 0 => Symmetric Oscillation.
QQ Q Q
AD, Phys. Rev. B 82, 172402 (2010).
Low 𝜔
QQ Q
The Peaks smooth out, and a monotonic behavior emerges AD, Phys. Rev. B 82, 172402 (2010).
Analytical Approach
Jordan-Wigner Transformation + Fourier Transform
A Bunch of Independent 2-level Problems:
No Exact Solution!
Probability vs Amplitude Flipping of Magnetization
↔
Population Dynamics
Probability Recursion
1
Every|𝑣𝑘 ||2 tends to 2 asymptotically independent of 𝜔 V. Mukherjee, A. Dutta and D. Sen, Phys. Rev. B 77, 214427 (2008).
Q → 0 independent of 𝝎
Solution for 𝝎 ≫ 𝑱: The RWA Unitary Transformation:
Transformed Wave-Function: Transformed Hamiltonian:
Jcos(k) << 𝜔 Highly Off-Resonant!
The Rotating Wave Approx (RWA): Keep only n = 0 S. Ashhab et. al., Phys. Rev. A 75 063414 (2007), AD, Phys. Rev. B 82, 172402 (2010).
“zero – photon process”
The Freezing of Response
Dynamical Many-body Freezing
Freezing Particle in a Double-Well “CDT”
2ℎ0 ) 𝜔
𝐽0 (
= 0 ⟹ |𝑣𝑘 (𝑡)|2 = |𝑣𝑘 (0)|2
Absolute Freezing: Every Mode Freezes !!
The Freezing is Independent of the Initial Condition. The effect is clearly visible even for Very Small system-sizes Possibility of Experimental Realization
AD, Phys. Rev. B 82, 172402 (2010). F. Grossman et. al., Phys. Rev. Letts. 57, 516 (1991).
The Q Formula With the Initial Condition:
Q
AD, Phys. Rev. B 82, 172402 (2010).
Q
DMF with Rectangular Pulse
Maximal Freezing for 𝚪𝟎 T = n𝝅 S. Bhattacharyya, AD and S. Dasgupta, Phys. Rev. B 86 054410 (2012).
Does Fast AC Driving Always Imply a DC ? An Interesting Finite Size Effect at the Freezing Peaks Sudden drops in Q for L = 2(2n+1) . Emergence of a Long but Finite Timescale. Recall:
AD and R. Moessner, arXiv:1208.0217v1
Full Oscillation for Cos(k) = 0 ..!
Can we see this drop in the L → ∞ limit? Can we have symmetric response (Q=0) for Arbitrary Fast Driving?
DMF in XY Chain The Hamiltonian:
We Define: Response (with Fully +z-Polarized Initial Condition):
𝟐𝒉
Absolute Freezing (𝑨𝒌 = 𝟎) for 𝑱𝟎 ( 𝟎 ) = 0 𝝎 Full Oscillation (𝑨𝒌 = 𝟏) for 𝜹 = 𝟎 !! AD and R. Moessner, arXiv:1208.0217v1
The DC Response Q
Q=
|𝛿| 𝛿 +|𝛾𝐽0 (2ℎ0 /𝜔)|
Pinch Point Discontinuity: The Limits 𝜹 → 𝟎 and 𝑱𝟎 (𝟐𝒉𝟎 /𝝎) → 𝟎 DO NOT COMMUTE Q Drops from 1 to 0 at 𝜹 = 𝟎
Total Slowing Down at the Discontinuity!
Massive Slowing Down of Every Mode at the pinch-point 𝜹 = 𝟎 and 𝑱𝟎 (𝟐𝒉𝟎 /𝝎) → 𝟎
𝑇𝑘 → ∞ for all k under RWA A Side Note: Simple-minded Single-Sweep Picture Fails!
AD and R. Moessner, arXiv:1208.0217v1
Experiment
S. Hegde, H. Katiar, T. Mahesh and AD (work in progress)
Thanks! To all of you for your patience! To my Collaborators: Sirshendu Bhattacharyya (R.R.R Mahavidyalaya, Hoogly) Subinay Das Gupta (Calcutta University, Kolkata) Roderich Moessner (Max Planck Institute for the Physics of Complex Systems, Dresden)
