Components of the neutral-atom machine

We experimentally immerse up to ten laser-cooled Cs atoms in the $$ \mid F_{\mathrm{Cs}}=3,m_{F,\mathrm{Cs}}=3⟩$$ state into an ultracold Rb gas of up to 10⁴ atoms in the state $$ \mid F_{\mathrm{Rb}}=1,m_{F,\mathrm{Rb}}=-1⟩$$, both species confined in a common optical dipole trap (Fig. 1a) (Methods A). Here F and m_F denote the total atomic angular momentum and its projection onto the quantization axis, respectively. The quantization axis is given by an external magnetic field of B₁ = 346.5 ± 0.2 mG or B₂ = 31.6 ± 0.1 mG. The Cs atoms quickly thermalize to the kinetic temperature of T = 950 ± 50 nK of the gas. We operate the quantum heat engine in the spin-state manifold of the seven Cs hyperfine ground states $$ \mid F_{\mathrm{Cs}}=3,m_{F,\mathrm{Cs}}⟩$$, m_F,Cs ∈ [+3, +2, …, −3], which define its quasi-spin. These states are energetically equally spaced with Zeeman energy $$ E_n^{\mathrm{Cs}}=n\lambda B$$, with $$ \lambda =\mid g_F^{\mathrm{Cs}}\mid {\mu }_{\mathrm{B}}$$, where $$ g_F^{\mathrm{Cs}}=-1/4$$ is the Cs Landé factor, μ_B Bohr’s magneton and n = 3 − m_F,Cs²⁰, with the zero-point of energy set to the lowest-energy state $$ \mid m_{F,\mathrm{Cs}}=3⟩$$.

Fig. 1

Operation principle of the quantum heat engine.

a Individual laser-cooled Cs atoms (green) are immersed in an ultracold Rb cloud (orange); both are confined in a common optical dipole trap (DT). External magnetic fields and microwave (MW) radiation, respectively, implement the power strokes of the quantum heat engine and distinguish the high- from the low-energy bath. The inset shows typical m_F-resolved fluorescence images of single Cs atoms for t = t_B = 300 ms after initialization, from which the quantized spin, and thus heat exchange, can be determined. The position of the bath cloud is indicated in orange with a width of 4σ. b The experimental Otto cycle consists of a heating stage, during which average heat 〈Q_H〉 is absorbed, and a power stroke induced by an adiabatic change of the magnetic field. A microwave field then switches the bath from high to low energy. The cycle is further completed by a cooling step, during which average heat 〈Q_C〉 is released, and an additional power stroke when the magnetic field is adiabatically brought back to its initial value. c The heat transfer between the Cs atom (engine) and a Rb (bath) atom occurs via inelastic spin-exchange collisions. In each collision, a single quantum of spin associated with a certain energy quantum is exchanged. Spin polarization of the Rb atoms and spin-conservation in individual collisions allow only up to six exo- or endothermal processes, corresponding to heating or cooling. d Owing to the difference of atomic Landé factors between Cs and Rb, the quantum heat engine (green) absorbs heat 〈Q_H〉 and releases heat 〈Q_C〉 (to produce work 〈W〉), while the bath releases more energy. The lost energy is irreversibly dissipated during an average of ten elastic collisions and is described by a heat leak 〈Q_L〉 from the high-energy bath.

Heat between the quantum engine and the bath is exchanged at the microscopic level via inelastic spin-exchange collisions (Fig. 1c). Each collision changes the value of the quasi-spin of the Cs engine by Δm_Cs = ∓1ℏ, leading to an energy change of ΔE^Cs = ±λB for each Cs atom, and Δm_Rb = ±1ℏ for one Rb atom corresponding to the energy change ΔE^Rb = ∓κB, with $$ \kappa =\mid g_F^{\mathrm{Rb}}\mid {\mu }_{\mathrm{B}}$$, where $$ g_F^{\mathrm{Rb}}=-1/2$$ is the Rb Landé factor¹⁴. The spin population thus directly reflects the energy exchange between engine and reservoir at the level of single energy quanta. The direction of the heat transfer is determined by the spin polarization of the Rb bath and by angular momentum conservation during individual collisions¹⁴. The spin polarization of the Rb atoms distinguishes a high-energy bath for m_Rb = −1 from a low-energy bath for m_Rb = +1. Control over the internal Rb state accordingly permits to either increase or decrease the energy of the quasi-spin of the engine. Heat exchange automatically stops after six spin-exchange collisions, because then the highest/lowest energy state has been reached. One collision transfers the colliding Rb atom to the $$ \mid F_{\mathrm{Rb}}=1,m_{F,\mathrm{Rb}}=0⟩$$ state, which forms the exhaust of the engine. Owing to the massive imbalance between the Rb and Cs atom numbers (N_Rb/N_Cs > 1000), the probability of a second collision with the same Rb atom is indeed vanishingly small. Memory effects are furthermore negligible, making the bath Markovian.

Implementation of the quantum Otto cycle

The quantum Otto cycle consists of four parts: one compression and one expansion step, during which work is performed, and a heating and a cooling stage, during which heat is exchanged¹³. The corresponding experimental sequence is shown in Fig. 1b. The Cs machine is first driven by up to six spin-exchange collisions into energetically higher states (at magnetic field B₁), absorbing average heat 〈Q_H〉 in time τ_H = t_B. Mean work 〈W_BC〉 is then performed by adiabatically decreasing the magnetic field to B₂ in τ = t_C − t_B = 10 ms. This time is much longer than the inverse energy splitting ΔE of the quasi-spin states, making the process adiabatic. It is, however, fast enough to avoid unwanted spin-exchange collisions, implying that no heat is transferred. The engine is subsequently brought into contact with the low-energy bath by flipping the spins of the Rb bath using microwave (MW) sweeps. The Cs engine is accordingly driven by up to six spin-exchange collisions into energetically lower states, releasing heat 〈Q_C〉 in time τ_C = t_D − t_C. Work 〈W_DA〉 is further performed by adiabatically increasing the magnetic field back to B₁ in τ = t_A − t_D = 10 ms. The Rb spins are finally flipped to their initial state with other microwave sweeps, restoring the high-energy bath.

While each single collision is coherent and thus amenable to quantum control¹⁵, coupling of the engine to the large number of bath modes in elastic collisions destroys the coherence between the engine’s quasi-spin levels. Heat is thus associated with changes of occupation probabilities, 〈Q〉 = ∑_nE_nΔp_n, whereas work corresponds to changes of energy levels, 〈W〉 = ∑_np_nΔE_n¹³. In our system, we concretely have $$ ⟨Q_{\mathrm{H}}⟩={\sum }_{n}n\left(p_n^{\mathrm{B}}-p_n^{\mathrm{A}}\right)\lambda B_1$$ for heating and $$ ⟨Q_{\mathrm{C}}⟩={\sum }_{n}n\left(p_n^{\mathrm{D}}-p_n^{\mathrm{C}}\right)\lambda B_2$$ for cooling. On the other hand, the respective work contributions for expansion and compression are given by $$ ⟨W_{\mathrm{BC}}⟩={\sum }_{n}n{p}_n^{\mathrm{B}}\lambda \left(B_2-B_1\right)$$ and $$ ⟨W_{\mathrm{DA}}⟩={\sum }_{n}n{p}_n^{\mathrm{D}}\lambda \left(B_1-B_2\right)$$. In order to evaluate these average quantities, we determine the magnetic fields B₁ and B₂ with the help of Rb microwave spectroscopy (Methods). We further detect the Zeeman populations $$ p_n^i$$ of single Cs atoms at arbitrary times by position resolved fluorescence measurements combined with Zeeman-state-selective operations (Fig. 1a inset)²¹. From each individual measurement, we can determine quantized spin transitions for each single atom. This allows us to monitor the resulting quantized heat exchange between engine and environment with a resolution of single quanta at each time. From a series of such measurements, we can further construct the average evolution of the quasi-spin populations (Fig. 2a, b): the progressive transfer from low (high) energy states to high (low) energy states during heating (cooling) as a function of time is clearly seen (green dots). From the measured heat counting statistics, we compute average (blue and red dots) and variance of heat exchange. We will use these quantities to examine the power output of the quantum machine and its fluctuations.

Fig. 2

Full-counting statistics of heat exchange.

During the heating (AB) and cooling (CD) steps of the quantum Otto cycle (center), heat is exchanged with the bath. The average population dynamics of the individual engine levels are shown in green. The mean heats, 〈Q_C〉 and 〈Q_H〉, extracted from the full-counting statistics are indicated for a cooling (blue) and b heating (red), as a function of the respective times τ_C and τ_H. Dots show the experimental data, solid lines are a prediction of a microscopic model (Methods). In both panels, the population dynamics shows the transition from an initially spin-polarized engine state via a state of many populated m_F levels to a spin-polarized state of the other extreme spin state. The inversion of an initially fully polarized population ($$ \mid m_{F,\mathrm{Cs}}=3⟩\leftrightarrow \mid m_{F,\mathrm{Cs}}=-3⟩$$) requires some hundreds of milliseconds. Error bars show statistical uncertainty of ± 1σ standard deviation.

Performance of the quantum heat engine

We first characterize the performance of the quantum Otto engine by evaluating its efficiency given by¹³,

Fig. 3

Performance of the quantum heat engine.

a Efficiency η, Eq. (2) (blue dots), and internal (dissipationless) efficiency η_int (green diamonds) for different cycle times; dashed lines indicate the respective expected values. b Average power output, Eq. (3) (blue dots: experimental data, red solid line: theoretical model), with maximal value reached after almost 12 spin exchange collisions. c Fano factor, Eq. (4), and time-resolved fluctuations σ_P (inset). In all cases, the dashed vertical lines (upper axis) indicate the number of spin-exchange collisions N_SE. The different durations between two successive spin-exchange collisions originate from different atomic transition rates¹⁴. Error bars show statistical uncertainty of ± 1σ standard deviation.

Second, we consider the average power of the quantum heat engine which reads,

We finally investigate the stability of the quantum Otto engine by analyzing the relative power fluctuations via the Fano factor, which quantifies the deviation from a Poisson distribution²³,

Experimental procedures

We start our experimental sequence by preparing an ultracold Rb gas in the magnetic field insensitive state $$ \mid F_{\mathrm{Rb}}=1,m_{F,\mathrm{Rb}}=0⟩$$ and, at a distance of ≈200 μm, a small sample of laser-cooled Cs atoms. The Cs atoms are further cooled and optically pumped into the $$ \mid F_{\mathrm{Cs}}=3,m_{F,\mathrm{Cs}}=3⟩$$ hyperfine ground state by employing degenerate Raman sideband-cooling³⁰. A species-selective optical lattice³¹ transports the Cs atoms into the Rb cloud. MW radiation prepares the bath atoms in the state $$ \mid F_{\mathrm{Rb}}=1,m_{F,\mathrm{Rb}}=-1⟩$$. The starting point of the heat engine cycle is defined by switching off the optical lattice potential. After a predefined time t_i, the Cs-Rb interaction is stopped by freezing the positions of the Cs atoms using the optical lattice, and pushing the Rb cloud out of the trap with a resonant laser pulse. State-selective fluorescence imaging of the Cs atoms completes the procedure³².

The high-energy and low-energy baths are interchanged by transferring the Rb atoms from $$ \mid F_{\mathrm{Rb}}=1,m_{F,\mathrm{Rb}}=-1⟩$$ to $$ \mid F_{\mathrm{Rb}}=1,m_{F,\mathrm{Rb}}=+1⟩$$ and vice versa using two successive Landau–Zener sweeps. The transfer takes ~4.4 ms, which is fast enough to avoid spin-exchange interactions during the state change of the bath. The two magnetic fields B₁ and B₂ defining the quantization axis for the engine operation, are measured using Rb microwave spectroscopy on the $$ \mid F_{\mathrm{Rb}}=1,m_{F,\mathrm{Rb}}=0⟩\to \mid F_{\mathrm{Rb}}=2,m_{F,\mathrm{Rb}}=+1⟩$$ transition. The population of the Rb atoms in state $$ \mid F_{\mathrm{Rb}}=2,m_{F,\mathrm{Rb}}=+1⟩$$ is detected by standard absorption imaging, using a time-of-flight measurement (Fig. 4). We fit the measured data with a standard model to extract the transition frequency, which translates into a magnetic field value using the Breit-Rabi formula³³. We find typical errors of the order of 0.1 mG.

Fig. 4

Magnetic field extraction.

a Rb microwave spectra for extraction of the magnetic fields B₁ and B₂. Center illustrates the engine cycle and the corresponding Zeeman energy splitting of a Rb bath atom. Red lines correspond to the theory curves and blue dots are experimental data. These measurements yielding magnetic fields B₁ = 346.5 ± 0.2 mG and B₂ = 31.6 ± 0.1 mG. Measured spectra confirm similar magnetic fields for B and C. b Corresponding microwave transition scheme in the Rb ground-state hyperfine manifold.

The magnetic field changes extracting work of the engine have to be adiabatic, i.e., preserving the populations p_n. The adiabaticity condition writes $$ \dot{{\omega }_{\mathrm{lar}}}/{\omega }_{\mathrm{lar}}^2\ll 1$$, where $$ {\omega }_{\mathrm{lar}}=\mid g_F^{\mathrm{Rb}}\mid {\mu }_{\mathrm{B}}B/\hslash $$ is the Larmor frequency. It can, therefore, be expressed as

Experimentally, we linearly vary the magnetic field from B₁ = 346.5 ± 0.2 mG to B₂ = 31.6 ± 0.1 mG in a time scale of 10 ms, yielding values of A(B₁) = 0.2 × 10⁻³ and A(B₂) = 14 × 10⁻³, thus fulfilling the adiabatic condition at any time during the variation of the magnetic field. Moreover, the time scale of the magnetic field variation is faster than the time scale associated with the spin exchange collisions (see number of collisions over time in Fig. 3). Hence, the populations p_n are constant during the isentropic processes (B → C and D → A).

Microscopic model and number of collisions

The quantum heat exchange between engine and bath is based on the understanding of individual spin-exchange collisions. In general, the spin-collision rate $$ {\mathrm{\Gamma }}^{m_F\to m_F\pm 1}$$ is different both for every initial state m_F and for the direction, i.e., Δm_F = ±1. The individual rates are well known from coupled-channel calculations of the molecular interaction potential between Rb and Cs¹⁴. These rates allow us to describe the evolution with a rate model²¹ that captures the spin dynamics and yields excellent agreement with the experimental data. From these rates, we also compute the mean number of spin collisions N_spin within a cycle duration t = t_D in two steps. First, we calculate the time-averaged collision rate as the sum of time-averaged collision rates during heating (exothermal spin collisions) and cooling (endothermal spin collisions) as

Second, we integrate these rates during the heating and cooling to obtain the number of collisions within cycle time t as

In order to close the cycle, the inital and final Cs states before and after a cycle have to be the equal, leading to the condition N_A→B = N_C→D.

Efficiency of the endoreversibe machine

We calculate the efficiency by distinguishing two different forms of heat exchange. First, we consider the respective mean energies given (〈Q₁〉) and taken (〈Q₂〉) by the baths, where 〈Q₁〉 − ∣〈Q₂〉∣ is the energy turnover of the reservoirs per cycle. Second, we consider the average energies absorbed (〈Q_H〉) and rejected (〈Q_C〉) from the engine, where 〈Q_H〉 − ∣〈Q_C〉∣ is the energy turnover of the machine. Both quantities differ because of the different atomic Landé factors of Cs and Rb. The difference $$ ⟨Q_{\mathrm{L}}⟩=\left(⟨Q_1⟩-\mid ⟨Q_2⟩\mid \right)-\left(⟨Q_{\mathrm{H}}⟩-\mid ⟨Q_{\mathrm{C}}⟩\mid \right)$$ is dissipated via elastic collisions and irreversibly lost to the kinetic energy of Rb. We macroscopically model it as a heat leak from the high-energy reservoir. Using the population distribution of the quasi-spin levels at the cycle points in Fig. 2 of the main text, the individual heats can be calculated, leading to

Owing to preservation of populations during adiabatic strokes, we can further use $$ p_n^{\mathrm{D}}=p_n^{\mathrm{A}}$$ and $$ p_n^{\mathrm{B}}=p_n^{\mathrm{C}}$$, yielding the expression for the dissipated heat

The efficiency is calculated as the work, ∣〈W〉∣ = 〈Q_H〉 − ∣〈Q_C〉∣, produced by the engine, divided by the energy provided by the high-energy bath, 〈Q_H〉 + 〈Q_L〉. Using $$ p_n^{\mathrm{D}}=p_n^{\mathrm{A}}$$, $$ p_n^{\mathrm{B}}=p_n^{\mathrm{C}}$$ and γ = λ/κ, we find

The internal efficiency of the engine is computed as the ratio of the produced work ∣〈W〉∣ and the heat absorbed by the machine 〈Q_H〉:

It corresponds to the efficiency without a leak (γ = 1).

Fluctuations of the quantum machine

To extract the fluctuations of the engine, Eq. (4), we calculate the mean power, Eq. (3), via 〈P〉 = ∣〈W〉∣/τ_cycle. The cycle time τ_cycle = t_D is experimentally controlled, and we assume that it is a fixed parameter not adding further fluctuations to the power-output fluctuations. Therefore, we can restrict the calculation to the fluctuations σ_W of work 〈W〉 as $$ {\sigma }_W^2=⟨W^2⟩-{⟨W⟩}^2$$. The work is given by the difference of energy absorbed by and rejected from the engine ∣〈W〉∣ = 〈Q_H〉 − ∣〈Q_C〉∣, and hence

The averages and variances of heat absorbed or rejected depend on the energy differences at the different points during the cycle, for example, 〈Q_H〉 = E(t_B, B₁) − E₀(t₀, B₁). Here, $$ E\left(t_i,B_j\right)={\sum }_n{p}_n^i\left(t_i\right)n\lambda B_j$$ can be computed from the measured populations $$ \left\{p_n^i\right\}$$ of level n at point i = A, B, C, D during the cycle and the magnetic field B_j(j = 1, 2), together with mean energy and variance. Then, the fluctuations $$ {\sigma }_Q^2$$ of heat 〈Q〉 exchanged when changing the engine’s probability distribution from point i to point f at a magnetic field B_j reads