Rubidium atoms let physicists retrodict quantum spin beyond uncertainty

Using 5.17 x 10^(10) rubidium atoms, researchers from Shanxi University, Fudan University, and the Niels Bohr Institute have demonstrated the ability to retrodict quantum spin with precision exceeding the limits imposed by the Heisenberg uncertainty relation. The team monitored spin observables, achieving variances 1.01 dB below the established bound for over 1.4 hours. This work reveals that restrictions on predicting quantum measurements do not apply to retrodictive guessing of past measurements. This finding builds on the seminal work of Werner Heisenberg, who first articulated the uncertainty relation in 1927.

Rubidium Ensemble Enables Continuous Retrodiction of Quantum Spin

Using 5.17× ${10}^{10}$ rubidium atoms, this achievement allows physicists to infer the state of a quantum system in the past with greater precision than previously thought possible. This oscillator’s position and momentum operators, $x_A$ and $p_A$, were then monitored through weak, continuous measurements.

By analyzing measurement records obtained both before and after a specific time, t, the researchers were able to retrodict the outcomes of measurements at that time with a precision exceeding the bounds of the uncertainty relation. This process relies on a stochastic master equation that describes the conditional evolution of the spin-oscillator state, accounting for quantum back-action and environmental interactions.

This approach differs from previous work using quantum memory on multiple bits or similar retrodiction in spin systems, as it achieves concurrent sub-SQL variances with direct access to both observables via continuous weak measurement records. The ability to retrodict quantum states with such precision opens avenues for advanced quantum sensing techniques, potentially combining trajectory and entanglement strategies. The researchers highlight that their past quantum state formalism enables the reconstruction of Wigner functions for arbitrary quadratures of the spin oscillator, offering a more complete understanding of the system’s quantum state.

This finding builds on earlier demonstrations of retrodiction in optical systems and other spin systems, but distinguishes itself through its continuous nature and direct access to both key observables. The implications of this work extend beyond fundamental quantum mechanics, potentially impacting the development of more sensitive measurement devices and enhancing their ability to probe the quantum world.

Through weak continuous measurements, they monitor canonical spin observables $x_A$ and $p_A$ with [$x_A,p_A$] = iℏ, and retrodict the outcomes of measurements of $x_A$ and $p_A$ at time t with variances that remain, concurrently over 1.4 hours, ~ 1.01 dB below the bound set by the Heisenberg uncertainty relation. The researchers conclude that their approach offers a promising pathway for further research into quantum measurements and the development of quantum-enhanced technologies.

Heisenberg Uncertainty Relation and Standard Quantum Limit

The ability to circumvent a cornerstone of quantum mechanics, the Heisenberg uncertainty relation, has been demonstrated using a collective spin oscillator comprised of 5.17 x ${10}^{10}$ rubidium atoms. Researchers achieved variances in predicting quantum spin states that fall approximately 1.01 dB below the established limit dictated by the Heisenberg uncertainty relation, a feat previously theorized but not experimentally verified for simultaneous estimation of non-commuting variables.

This retrodictive precision wasn’t achieved through stronger measurement techniques, but by intelligently combining information gathered both before and after the moment of measurement. This work extends prior demonstrations of retrodiction, which had previously focused on mechanical oscillators and collective spin systems, by achieving simultaneous sub-standard quantum limit (SQL) retrodiction of arbitrary linear combinations of position and momentum quadratures.

The team’s approach uses the concept of a quantum trajectory, reconstructing the system’s evolution by continuously monitoring the $x_A$ and $p_A$ components of the spin oscillator, allowing for a detailed reconstruction of the system’s past state. This allows for a reduction in uncertainty beyond what is typically possible under the standard Heisenberg framework, effectively “squeezing” the quantum state to reveal more precise information about its past. This brings interesting prospects for further research on quantum measurements, and for quantum-enhanced sensing that combines trajectory and entanglement strategies.

Experimental Setup for Monitoring Spin Observables

Using a spin oscillator, researchers employed a stroboscopic technique to sequentially measure canonical spin observables $x_A$ and $p_A$, a method designed to intertwine and alternate measurements of these properties. A key element of the setup involved modulating probe light at four times the Larmor frequency, facilitating the simultaneous measurement of $x_A$ and $p_A$. The resulting data, while initially noisy due to unavoidable back-action noise from the interaction, underwent processing using the past quantum state formalism.

The experiment achieved conditional variances for both x and p, as well as for all quadratures in phase space, that consistently fell below the standard quantum limit. Measurements revealed an average noise of 0.388 ± 0.025, a significant reduction compared to the SQL of 0.5. To their knowledge, this is the first experimental demonstration of sub-SQL conditional estimation over all phase-space quadratures.

This continuous, sub-SQL performance distinguishes this work from previous demonstrations that achieved similar results with quantum memory using optical systems or in different spin systems. The measured atomic spin noise, scaled to atomic projection noise, showed a clear reduction in uncertainty when analyzed using the past quantum state formalism.

This setup’s success hinges on the combination of continuous weak measurements and the application of quantum trajectory theory, allowing for the concurrent estimation of non-commuting observables. The experiment’s design, with its focus on direct access to both observables and continuous monitoring, represents an advancement in the field of quantum metrology.

Continuous Weak Measurements of x_A and p_A

The precision of retrodicting quantum states has been demonstrably improved through an experiment using 5. This achievement relies on the continuous monitoring of non-commuting spin observables, $x_A$ and $p_A$ and a novel application of a hybrid classical-quantum theory initially proposed by Tsang for estimating classical Markov processes. Data collected over 1.4 hours revealed conditional variances approximately 1.01 dB below the Heisenberg uncertainty relation’s bound, a level of precision sustained through over 50,000 repeated measurements for each data point.

These continuous, non-demolition measurements were performed via Faraday rotation, employing a far-detuned probe laser to interact with the spin oscillator, governed by a Hamiltonian that allows for weak probing without significantly disturbing the system. The process involves alternating measurements of orthogonal quadratures, reducing the variance of one observable while introducing quantum back-action noise to the other, a technique refined through the integration of measurements over time.

This equation, coupled with the measurement records obtained during the probing process, allows for the reconstruction of the quantum state and the subsequent retrodiction of past measurements. The steady-state prediction variance, while slightly above the standard quantum limit due to imperfections in spin polarization, approaches 0.5 under ideal conditions, perfect pumping and quantum detection efficiency, mirroring the variance of a coherent spin state. The high precision achieved isn’t in determining joint projective measurement outcomes, but rather in the conditional estimation of both non-commuting observables reconstructed from the continuous weak measurement records.

Unlike projective measurements that destroy squeezing in the conjugate component due to quantum back-action, this method preserves information through weak measurements, even incorporating small back-action effects from alternating x and p measurement pulses. This work builds on prior experimental confirmations of the formalism using cavity QED, superconducting qubits and other atomic ensembles, demonstrating its broad applicability.

Retrodiction Achieves Variances Below Uncertainty Bound

Researchers were able to surpass a fundamental quantum limit, achieving variances in retrodicted measurements 1.01 dB below the Heisenberg uncertainty relation for over 1.4 hours. This demonstration of sub-SQL (standard quantum limit) variances, simultaneously for both non-commuting quadratures, marks a first for steady-state spin oscillators and relies on a technique combining prediction with retrodiction.

Prior work demonstrated retrodiction’s potential, but those studies exhibited larger uncertainties in linear combinations of measured quantities, a limitation now overcome through this new approach. This achievement hinges on the retrodiction being conditioned on measurement results obtained both before and after the time of interest; single-sided estimation, relying only on prior results, yielded variances at best equal to the SQL.

The experimental setup’s ability to access both observables through continuous weak measurement records distinguishes it from previous demonstrations. Earlier work achieved similar retrodiction in spin systems and optical systems, but those experiments did not achieve concurrent sub-SQL variances.

The red and light red curves in their data represent variances exceeding the SQL by 4.03 dB without conditioning, while the green and light green curves show variances 0.88 dB above the SQL using prior measurements. The team’s success demonstrates the power of combining forward prediction with retrodiction within the past quantum state formalism, opening avenues for quantum-enhanced sensing strategies that integrate trajectory and entanglement techniques.

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