Deterministic Squeezing Regime in Bose-Hubbard Systems

• Deterministic squeezing regime is a quantum evolution where noise reduction is achieved via Hamiltonian-controlled parameter ramps rather than stochastic measurements.
• It utilizes the Bose-Hubbard double-well model, enabling a quasi-adiabatic transition through the Rabi, Josephson, and Fock regimes to generate strong number squeezing.
• The process reveals critical quantum many-body effects, including parity sensitivity and the interplay between classical and quantum evolution, offering robust applications in precision metrology.

A deterministic squeezing regime is a dynamical or engineered quantum evolution in which the squeezing (reduction of quantum noise in a collective or single-mode observable) emerges in a manner that is unconditional, reproducible, and set by the system Hamiltonian or controlled parameter ramps, rather than by stochastic measurement outcomes or post-selection. This regime contrasts sharply with conditional or measurement-based squeezing, in which the noise reduction depends on the outcomes of continuous quantum non-demolition (QND) measurements or feedback based on those records. Deterministic squeezing protocols are of particular interest because they enable robust, on-demand preparation of squeezed quantum states for use in metrology, quantum information, and fundamental studies.

1. Theoretical Framework and Model System

The deterministic squeezing regime described in "Two-mode Bose gas: Beyond classical squeezing" (Bodet et al., 2010) is analyzed in the context of a two-mode (double-well) ultracold Bose gas governed by the two-site Bose–Hubbard Hamiltonian: $$\hat{H} = -J(\hat{a}_1^\dagger \hat{a}_2 + \hat{a}_2^\dagger \hat{a}_1) + \frac{U}{2}\sum_{i=1,2} \hat{a}_i^\dagger \hat{a}_i^\dagger \hat{a}_i \hat{a}_i,$$ where $J$ is the tunneling rate and $U$ the onsite interaction. Reformulating in Schwinger angular-momentum operators: $$S_1 = (\hat{a}_2^\dagger \hat{a}_1 + \hat{a}_1^\dagger \hat{a}_2)/2, \quad S_2 = i(\hat{a}_2^\dagger \hat{a}_1 - \hat{a}_1^\dagger \hat{a}_2)/2, \quad S_3 = (\hat{a}_1^\dagger \hat{a}_1 - \hat{a}_2^\dagger \hat{a}_2)/2,$$ the Hamiltonian takes the form

$\hat{H} = -J (S_+ + S_-) + U(S_3^2 + S^2 - N).$

Number squeezing is quantified by the dimensionless parameter

$$\xi_3 = \frac{4(\Delta n)^2}{N} = \frac{4(\Delta S_3)^2}{N},$$

which compares the variance $(\Delta S_3)^2$ of the number difference to the quantum shot noise level.

2. Deterministic Squeezing via Hamiltonian Control

The deterministic squeezing regime is accessed by dynamically and quasi-adiabatically changing (ramping) the height of the potential barrier between the wells. The process is as follows:

• At low barrier, the Bose gas forms a delocalized angular–momentum coherent state (minimum uncertainty, $\xi_3\approx 1$).
• As the barrier is slowly increased, $J$ decreases (tunneling suppression) and $U$ remains constant or increases, driving the system out of the Rabi regime, through the Josephson regime, and into the Fock regime.
• During the ramp, the many-body state follows a deterministic trajectory in the $(\alpha, \Delta n)$ plane, where $\alpha = 2\langle S_1 \rangle/N$ measures coherence, guided by the instantaneous Hamiltonian.
• The evolution is subject to the Heisenberg uncertainty constraint,

$$(\Delta S_k)^2 (\Delta S_l)^2 \geq \frac{1}{4} |\epsilon_{klm} \langle S_m \rangle|^2,$$

which bounds possible squeezing.

For sufficiently slow ramps, the system "deterministically" approaches minimal number fluctuations at fixed coherence, nearly reaching the lower quantum limit for $\xi_3$ over a wide $\alpha$ range. Fast ramps freeze-in the squeezing once the evolution timescale becomes short compared to the instantaneous Josephson plasma frequency,

$\omega_p = \sqrt{2J(NU + 2J)},$

with a frozen squeezing value

$$\xi_3 \approx \sqrt{1 + \left(\frac{\pi N U \tau}{2}\right)^2} - \frac{\pi N U \tau}{2},$$

where $\tau$ is the effective ramp timescale.

3. Transition between Josephson and Fock Regimes

The nature of squeezing strongly depends on the ratio $U/J$:

• Josephson regime: $N^{-1} \ll U/J \ll N$. Moderate number squeezing with high phase coherence; plasma-like linear excitation spectrum.
• Fock regime: $U/J \gg N$. Interactions dominate, tunneling is negligible, and low-lying eigenstates become quasi-degenerate pairs with very strong number squeezing ($\xi_3 \ll 1$). However, the relative phase of the two modes becomes undefined.

In the Fock regime, the ground state is a symmetric (even) superposition centered at $n=0$ (for even total atom number), exhibiting nonclassical correlations not accessible via mean-field theory or classical simulations.

4. Quantum Many-Body Effects: Parity Sensitivity

A key quantum feature emerging in the deterministic squeezing regime is the sensitivity to the parity of the total atom number $N$:

• Even $N$: The ground state can be a "cat-like" symmetric superposition, enabling very low (sub-shot-noise) number fluctuations.
• Odd $N$: Symmetry forces one particle imbalance, so $\xi_3$ cannot fall below the classical limit, despite the same classical mean-field trajectory.

Thus, precise measurement of relative number fluctuations can discriminate even and odd $N$ at the quantum level, an effect that is only present for systems deeply in the Fock regime and cannot be recovered by classical statistical approaches.

5. Quantum Versus Classical Evolution

The deterministic squeezing trajectory can be simulated in two ways:

• Quantum evolution: Solve the full von Neumann equation for the system's density matrix, tracking all quantum correlations.
• Classical statistical (semiclassical) evolution: Propagate an initial Wigner distribution via the classical equations of motion derived from the Bose-Hubbard Hamiltonian.

In regimes with large occupation numbers, both approaches yield similar evolution for $\xi_3$ and $\alpha$, indicating the squeezing formation is predominantly classical. However, at low temperatures and near the Fock regime, quantum effects (oscillatory features, negative Wigner function values, parity sensitivity) appear, highlighting the role of genuine non-classicality and the necessity of full quantum treatment for deterministic squeezing beyond classical limits.

6. Implementation Considerations, Performance, and Scaling

Deterministic squeezing by adiabatic barrier ramping imposes constraints on ramp speed (set by $\omega_p$), the ability to control $U/J$ over orders of magnitude, and the requirement to avoid decoherence and thermal effects during the slow ramp. The achievable squeezing ($\xi_3$) is limited by both the initial temperature, the total atom number $N$, the final $U/J$ ratio, and the ramp profile. The parity effect (even/odd $N$) requires single-atom resolution and ultra-low temperature. The controlled, Hamiltonian-driven trajectory enables robust and repeatable squeezing—in contrast to measurement-based, conditional, or feedback protocols which typically result in run-to-run fluctuations tied to the measurement record.

7. Significance and Application

Deterministic squeezing realized via slow parameter (barrier) ramps in ultracold double-well systems provides a direct, Hamiltonian-engineered path to strong nonclassical number fluctuations, with relevance to precision interferometry, quantum metrology (beyond the standard quantum limit), and fundamental studies of quantum correlations in mesoscopic many-body systems (Bodet et al., 2010). The scheme demonstrates how tailored, quasi-adiabatic control can deterministically drive a system to highly squeezed states that—deep in the quantum regime—display features not reproducible by semiclassical approximations. This approach sets a benchmark for deterministic, Hamiltonian-based squeezing protocols in other platforms.