Nondestructively probing the thermodynamics of quantum systems with qumodes

Quantum systems are now routinely created, manipulated, controlled and studied in the lab, making the realization of technologies that exploit truly quantum phenomena a very imminent reality. Examples of such systems are numerous, including ultracold atoms,¹ ion traps,² superconducting circuits,³ and microwave cavities.⁴ Beyond the intrinsic interest in studying such systems, they can be deployed to, for example, enhance computation and metrology.^5–8 A particularly promising use, especially in the near term, is that of analogue quantum simulation, where the system is made to emulate the behavior of another quantum system of interest,^9–11 which can be applied in quantum chemistry¹² to study otherwise experimentally inaccessible quantum systems and parameter regimes.

A necessary element in these applications is to read-out properties of or results output by the quantum system. While theoretical models of measurement are well-established, in many of the proposed architectures the measurement process is destructive to the system, allowing only a single-shot reading, after which the system must be completely reprepared. For example, with cold atom systems, a widely used measurement method is time-of-flight,^13–16 which involves destruction of the atomic trapping potential, hence requiring the atoms to then be re-trapped and re-cooled. Moreover, in other systems direct methods to measure particular sets of observables may not be available.

It has been proposed that this can be mitigated through the use of ancillary systems that serve as measurement probes. Through their interaction, properties of the system are imprinted onto the probes,^17,18 and information about the system may then be obtained through measurement of the ancillary probes alone.^19–32 While such measurements disturb the system state and are hence not non-demolition, they do not destroy the system. They thus constitute a practical method to determine certain system properties whilst leaving the system intact. This method has been explored in particular for cold atom systems, where atomic impurities form ancilla qubits;³³ post-processing of the impurity measurement statistics then allows properties such as density²⁹ and temperature^24,30 to be determined.

In parallel, developments in continuous variable quantum information processing,^34–38 based on continuous variable quantum modes (qumodes) rather than qubits, provide new applications for quantum optics and collective atomic phenomena in quantum technologies. One recently proposed model of quantum computation uses squeezed qumodes as a resource for phase estimation of an operator.³⁹ 

Bringing these two themes together, qumodes offer a promising, flexible means of nondestructively probing quantum systems. With an appropriate initial qumode state, the statistics of the system operator to which the qumode couples can be mapped directly onto the qumode state. Subsequent measurement of the qumode then allows for the spectrum of the system operator to be determined, along with the populations of the respective eigenstates. This enables a full characterization of the moments of the operator for the system state, as though one had directly sampled the observable from the system. In the ideal case, the qumode behaves akin to a nondestructive von Neumann measurement meter,^40,41 while even with a more realistic model of achievable initial qumode states a highly faithful measurement can be performed. Crucially, we only directly measure the qumode ancilla, from which we are able to indirectly probe the system of interest itself, circumventing the typical direct approaches that are often destructive to the system.

Here, we highlight how we can employ these probes to study the thermodynamics of quantum systems. We first describe the basic qumode probing protocol, and how the operator statistics are imprinted onto the qumode state. We show that the protocol is robust to experimental limitations of finite squeezing in the initial qumode state, and discuss its experimental practicality. We then introduce several thermodynamical applications of the probing protocol. Amongst these, we show how qumodes can be used as thermometers to measure the temperature of quantum systems in equilibrium, and how partition functions can be reconstructed. We also show how reversible and irreversible contributions to work can be measured in non-equilibrium settings, extending beyond prior work that introduced an analogous approach to measuring total work.^42,43 We also show how the protocol can be used to measure the overlaps of ground states of different parameter regimes of a Hamiltonian. As an example, we simulate an illustrative use of our method in probing the temperature of a spin-1/2 system in a transverse field.

We first describe the basic qumode probing protocol, from an architecture-agnostic perspective. Consider a quantum system of interest with (possibly unknown) state $ρ sys$⁠, and an observable $O$ of the system that we wish to measure. Specifically, we wish to determine the moments of $O$ with respect to $ρ sys$⁠, given by $⟨ O m ⟩ = Tr ( ρ sys O m )$⁠, where O is the operator associated with $O$⁠. The aim is to make these measurements in a manner nondestructive to the system. This can be achieved by a probing protocol that uses ancillary qumodes as probes. We emphasize that the protocol is not a full tomography of the state or operator,^44–47 but rather, a means to obtain the statistics of the observable with respect to the system state.

The qumode probing protocol consists of three components (see Fig. 1). The first of these is the system of interest, with state $ρ sys$⁠. The protocol does not in general need a particular form for the system or its state, and it may inhabit a discrete or continuous Hilbert space. As mentioned above, there is no need for a priori knowledge of this state, and in general this nescience is assumed. The second component is the continuous variable qumode, described by its quadratures x and p, often referred to as “position” and “momentum.”⁴⁸ We shall take these quadratures to be in their dimensionless form; that is, in terms of the creation and annihilation operators a and $a †$ of the mode, we have $x = ( a + a † ) / 2$ and $p = ( a − a † ) / 2 i$⁠.

The final component is the interaction between system and qumode. We shall consider an interaction Hamiltonian of the form $λ x ⊗ H int$⁠, where the first subspace belongs to the qumode, and the second the system.³⁹ Hence, the interaction acts on the system, with a strength that depends on the qumode position quadrature, with λ an overall coupling strength.⁴⁹ The associated evolution operator (in natural units $ℏ = 1$⁠) is $U ( t ) = exp ( − i λ x ⊗ H int t )$⁠, and thus the qumode is dephased in this quadrature, at a rate dependent on the system operator $H int$⁠, thence motivating the use of a phase estimation-type algorithm.

Before discussing the choice of initial probe state distribution G(x), let us address a caveat to the above derivation. Namely, that we have neglected the presence of the natural evolution of the system under its bare Hamiltonian H₀ during the running of the protocol. For this to be valid, we require that the interaction occurs on timescales much faster than the natural evolution (⁠ $λ H int ≫ H 0$⁠), and that the natural evolution has negligible effect on the system state during the running of the protocol (⁠ $H 0 τ ≪ 1$⁠), hence imposing a maximum allowable running time for the protocol. However, these restrictions are lifted when the bare Hamiltonian and the interaction Hamiltonian commute, in which case the natural evolution does not affect the outcome of the qumode measurement.

Indeed, the form of this distribution should not be too surprising—the smearing of the initial momentum distribution manifests in a smearing in the final distribution. This smearing is identical in the initial and final distributions, and is controlled by the squeezing factor. Specifically, the standard deviation of the final momentum distribution is given by $ς p = 2 / s$⁠, corresponding to a standard deviation for the system operator eigenvalues of $ς E = 2 / s λ τ$⁠. Thus, the precision to which we can measure the eigenvalues of $H int$ can be increased by running the protocol for longer or increasing the coupling strength, or increasing the squeezing. This is analogous to a result found in the proposal for the power of one qumode computation,³⁹ that one can trade-off a decreased squeezing with increased running time τ, and vice versa.

While it is tempting to then conclude that these uncertainties in the final distribution can thus be negated by a sufficiently increased running time, this is not necessarily practical in general, due to the constraint imposed on τ for the effects of the system's natural evolution H₀ to be neglected—as well as the inherent difficulties with maintaining coherence over long timescales. Ultimately, such practical considerations bound how small the spread $ς E$ can be made.

More generally, suppose we express the initial qumode state in the momentum eigenbasis as $| ψ q ⟩ = ∫ d p f ( p ) | p ⟩$⁠. We note that f(p) and g(x) are related by a basis change corresponding to a Fourier transform, i.e., $f ( p ) = G ( p )$⁠. Thus, we can equivalently express Eq. (4) as $P ( p ) = ∑ n P n | f ( p + λ E n τ ) | 2$⁠; the final distribution is a mixture of shifted versions of the initial momentum distribution, with weights given by the probability of each eigenstate of $H int$⁠, and shifts proportional to the corresponding eigenvalue.

Both cavity^51,52 and circuit^53–56 quantum electrodynamics experiments consist of interactions between a continuous variable mode (cavity fields in the former, nanomechanical resonators in the latter) and a two-level system (atoms and superconducting qubits, respectively), interacting through a quantum Rabi Hamiltonian Eq. (8) in the (ultra)strong coupling regime. Such setups operate in a regime where the qumode measurement resolution can be much finer than the differences between the interaction Hamiltonian eigenvalues, which for this example is of order unity. For example, in Ref. 54, the coupling between qubit and resonator gives $λ ≈ 10 10$⁠, and the Q-factor of 10³ and resonance frequency of 8.2 GHz leads to $λ τ ∼ 200$ when the protocol is run for times of the order of the resonator lifetime. With the parameters of Ref. 52, we would have $λ τ = 40$ when τ is the cavity lifetime. Comparing with our example later, we see that this would offer a high degree of precision, with a very fine resolution of the spectrum.

While Eq. (8) makes it clear that the protocol can be used to probe moments of the spin operator of a two-level system, it can be applied more generally. First, the physical motivation and derivation of the Hamiltonian does not necessarily require that the system has only two states, and can be rederived for any number of states, by replacing σ_x with the appropriate spin operator for the number of states. Second, by illuminating an array of such systems with the same light field, the Hamiltonian becomes an interaction between the light field and the sum of the individual spin operators for each system, and thus probes moments of the total spin operator, as well as correlations between individual spins. This can be enhanced by tuning the optical geometry⁵⁷ such that different spins couple to the qumode with different strengths and phases. Further, when the individual spins are indistinguishable, they couple to the qumode as a single, collective spin operator, leading to the Dicke model $H D = λ x ⊗ J x$⁠,^58,59 which may be probed in a similar manner. Finally, higher-order light–matter interactions in the context of many-body systems often appear in such a form,⁶⁰ such as in the case of cold atoms in optical cavities.^61,62

We emphasize that the “momentum” measurement made of the qumode is that of its canonical momentum quadrature, which does not necessarily correspond to a Newtonian momentum. Measurement of such quadratures is a highly routine procedure for photons in quantum optics,⁶³ and can similarly performed for phonon qumodes formed by e.g., atomic or ionic motion.^42,64

Consider a system that is known to be in a state of thermal equilibrium $ρ Θ ( β ) = exp ( − β H Θ ) / Z ( β )$ with respect to a Hamiltonian $H Θ$⁠. Here, $β = 1 / T$ is the inverse temperature (we employ units in which Boltzmann's constant k_B = 1), and $Z ( β ) = Tr ( exp ( − β H Θ ) )$ is the partition function. When the system component of the qumode interaction Hamiltonian $H int$ is proportional to $H Θ$⁠,⁶⁵ we can use the qumode as a thermodynamical probe of the state.

Unlike traditional means of thermometry,⁶⁶ we do not require thermal equilibrium to be established between the system and the probe—the temperature is measured through the shifts in the qumode state. Further, while similar schemes for measuring temperature using impurity probe qubits^24,30 also do not require equilibriation, their accuracy is limited by the number of probes. This can be a limitation even for quantum thermometers that exploit advantages in precision using quantum metrology.^24,67 In contrast, the precision of qumodes can also be enhanced by either increasing the squeezing factor, or increasing the interaction time or strength.

This approach requires that we can resolve between different eigenvalues of the interaction Hamiltonian ${ E n }$⁠, i.e., min(⁠ $| E n − E n ′ | ) ≳ ς E$⁠, or at least for the two eigenvalues used in Eq. (10) and their nearest neighbors. Otherwise, we must combine nearby eigenvalues and treat them as degenerate, in term limiting the precision to which the temperature T can be estimated by the (scaled) uncertainty $ς E ¯$⁠. The number of measurements required to determine the temperature to a given precision will scale inversely proportional to both the variance of the measurements of E_n (set by s, λ, and τ) due to the finite width of the output distribution peaks, and the associated eigenstate probabilities $P n 0$ and $P n 1$ involved in Eq. (10).

Particularly valuable is the access reconstruction of $Z ( β )$ provides to the free energy landscape of the system. Previously, proposals have been introduced, based either on the two-time measurement method, or using a qudit-⁷¹ or qumode-based^42,43 method, to sample from the distribution of work done on a system P(W) between two timepoints, where W corresponds to the difference in energy before and after the evolution (indeed, the latter of these can be seen as an important special case application of our qumode-probing framework). Then, using the quantum counterpart^72,73 of the Jaryzinski equality,⁷⁴ the free energy difference is deduced from the work done— $⟨ exp ( − β W ) ⟩ = exp ( − β Δ F )$⁠. However, these methods are not always efficient, particularly at low temperatures, or when large negative values of work are involved.⁷¹ In contrast, our method based on probing ${ E n }$ and ${ P n }$ directly still allows for the free energy to be recovered efficiently in those regimes.

Qumodes can also be used to probe the thermodynamics of quantum systems that are perturbed far from equilibrium. In particular, we can probe the average work performed on a system due to a sudden quench in the interaction Hamiltonian—and moreover, determine the irreversible portion of this work $⟨ W irr ⟩$⁠,^72,75,76 defined as the difference between $⟨ W ⟩$⁠, average work done on the system during the quench, and $Δ F$⁠, the change in the free energy had the system evolved adiabatically from the thermal state of the initial interaction Hamiltonian to that of the final Hamiltonian. Interest in the irreversible component of work is motivated by the fluctuation theorems and its connections with various entropy measures^72,77—it is a widely used measure of irreversibility, and has been shown to be a signature for some second-order phase transitions.⁷⁸ Note that the average work is also an interesting quantity to study in its own right, and its behavior across a critical point has been connected to first-order quantum phase transitions.⁷⁸ 

The means of measuring average work and its irreversible component proceeeds as follows. Consider initial and final interaction Hamiltonians $H int ( 0 )$ and $H int ( 1 )$⁠, the former proportional to the system Hamiltonian $H Θ$⁠, and the latter satisfying $λ H int ( 1 ) ≫ H Θ$⁠. For each of the interaction Hamiltonians we can apply the methods above to determine the free energies of their respective associated thermal states $ρ Θ i$⁠, and correspondingly, the difference in these free energies $Δ F$⁠. Under a sudden quench, the system is unchanged and will remain in its initial thermal state $ρ Θ$⁠, which is equivalent to a thermal state of $H int ( 0 )$ at some effective inverse temperature $β ̃$⁠. The proportionality between the system Hamiltonian $H Θ$⁠, and the initial interaction Hamiltonian $H int ( 0 )$ corresponds to the ratio between their eigenvalues $E n / E ¯ n$⁠, which also corresponds to the ratio of their associated temperatures $T ̃ / T = β / β ̃$⁠.

Finally, we remark that we can also use qumode probes to find the overlaps of the ground states of a parameter-dependent Hamiltonian at two different values of the parameter γ. Such overlaps have been used to characterize regions of criticality defining quantum phase transitions, such as in the Dicke model.⁷⁹ Let $H int ( γ )$ denote the Hamiltonian when the parameter takes on a particular value γ, and let γ_c be the value of the parameter at the critical point. The overlap of the ground states $| u γ ⟩$ associated with two values of the parameter $γ 0 < γ c$ and $γ 1 > γ c$ can be found using a concatenation of two qumode probing circuits. The first probing circuit is run using $H int ( γ 0 )$⁠, and is post-selected on the qumode measurement resulting in the initial ground state energy, such that the state is left in the ground state of this Hamiltonian $| u γ 0 ⟩$⁠. We then probe this state using a second qumode, coupled with the interaction Hamiltonian $H int ( γ 1 )$⁠. The probability that the qumode is found to have been shifted according to the ground state of this latter Hamiltonian then corresponds to $| ⟨ u γ 0 | u γ 1 ⟩ | 2$⁠, i.e., the desired overlap probability.

We now illustrate thermometry with a qumode by applying it to this example, when the system is thermalized to this Hamiltonian at a range of temperatures T. Specifically, we simulate sampling from the distribution obtained by employing the probing protocol with a qumode initially prepared in a squeezed momentum state. We study a scenario where the interaction Hamiltonian is proportional to the system Hamiltonian, i.e., $H int ∝ H sys$⁠.

We will also, as an extra test of the protocol, mimic that the eigenvalues of $H sys$ and $H int$ are unknown (but their relative proportionality is), and demand that the protocol also deduce these. We assume that it is known that there is one positive and one negative energy eigenvalue, such that the protocol will treat all measurements corresponding to a positive shift in momentum as being due to the negative eigenvalue, and vice versa. These are then averaged over to arrive at the deduced energy eigenvalues.

We examine the case that $Δ = B = 1 , H int = H sys$⁠. We consider temperatures in the range $β = 1 / T ∈ [ 0 , 2 ]$⁠, with the extremes corresponding to the infinite temperature limit (P₀ = P₁), and a very low temperature limit (⁠ $P 0 ≈ 0.997 , P 1 ≈ 0.003$⁠). We allow the qumode to make 10³ measurements (shots) of the distribution, and combine the dependence on squeezing s, coupling strength λ and interaction time τ in a single precision factor $α : = s λ τ$⁠. To illustrate the effect of the precision, we plot the final distributions of the qumode state for $α ∈ { 0.1 , 0.5 , 1 , 3 , 10 }$ at $β = 0.3$ in Figs. 3(a), 3(c), 3(e), 3(g), and 3(i), respectively.

The performance of the qumode at estimating the temperature of the spin for the prescibed range of temperatures is shown in Figs. 3(b), 3(d), 3(f), 3(h), and 3(j) for each of the respective precisions. We see that at low precision the method overestimates the temperature. This is because the shift of the qumode momentum during the interaction is much smaller than the finite width of the initial distribution, and so a significant portion of the tails of the distribution are erroneously assigned to the wrong eigenvalue, creating the illusion of a more balanced population of the two states. We also observe that generally the accuracy is lower at high temperature (low β); this is because the exponential dependence of the state probabilities on β renders a high degree of sensitivity in the deduced β with respect to the measured probabilities at high temperature (where $P 0 ≈ P 1$⁠). Indeed, when the measured probability for the excited state is greater than 1/2 (even if only to an arbitrarily small degree), the deduced temperature is negative. Ultimately, this can be ameliorated by using a larger number of shots that reduce the magnitude of the fluctuations in the measured probabilities. We remark that further numerical study (not shown) indicates that the number of shots is the limiting factor in the accuracy (rather than precision) for the shown plots at larger precision, and a visibly tighter clustering around the exact temperatues is observed for the measured temperatures at α = 10 when the number of shots is increased to 10⁴.

Properties of quantum systems may be imprinted onto continuous variable qumode ancillae, allowing for nondestructive probing of the system. For an appropriate choice of interaction operator and initial qumode state, the occupation probabilities of the desired observable's eigenstates with respect to the system state are mapped directly on to the qumode state. The qumode state then behaves according to the same statistics as this operator, and thus measurement of the qumode reproduces the same result as a direct measurement of the system would, while avoiding particular drawbacks associated with direct measurements of practical implementations of quantum technologies. We have shown that this method of probing has a strong potential for use in probing the thermodynamics of quantum systems, allowing access to many quantities of interest, particularly in the context of cold atom quantum simulators.

For systems in equilibrium, qumode probing can be used for thermometry of the system temperature, as well as reconstruction of the partition function, from which many other thermodynamical quantities of interest can be deduced. Moreover, for systems outside of equilibrium, qumodes can be used to probe work and free energy differences under evolutions of the system and its Hamiltonian. As the basic form of the interaction is almost synonymous with the form of quantum light–matter interactions, there is a significant scope of applicability of these results, particularly given that the necessary parameter regimes can be achieved in current experiments. Indeed, we remark that while several of our results require the system component of the interaction Hamiltonian to be proportional to the thermalizing Hamiltonian of the system, this is not an unrealistic challenge particularly in systems driven by their interactions with light or other similar environments, especially in the strong-coupling regime. With the high degree of tunability of the specific form of the interaction offered by optical setups, qumode probing promises to be a valuable tool in the experimental characterization of many-body quantum systems.

The authors acknowledge financial support from Grant Nos. FQXi-RFP-1809 and FQXi-RFP-IPW-1903 from the Foundational Questions Institute and Fetzer Franklin Fund (a donor advised fund of Silicon Valley Community Foundation), the University of Manchester Dame Kathleen Ollerenshaw Fellowship, the Imperial College Borland Fellowship in Mathematics, the Singapore Ministry of Education Tier 2 Grant No. MOE-T2EP50221-0005, the Singapore Ministry of Education Tier 1 Grant No. RG77/22, and the National Research Foundation, Singapore, and Agency for Science, Technology and Research (A*STAR) under its QEP2.0 programme (No. NRF2021-QEP2-02-P06), and funding from the Science and Technology Program of Shanghai, China (No. 21JC1402900). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not reflect the views of National Research Foundation or the Ministry of Education Singapore. T.J.E. and N.L. thank the Centre for Quantum Technologies for their hospitality.

The authors have no conflicts to disclose.

Thomas J. Elliott: Writing – original draft (equal); Writing – review & editing (equal). Mile Gu: Writing – original draft (supporting); Writing – review & editing (supporting). Jayne Elizabeth Thompson: Writing – original draft (supporting); Writing – review & editing (supporting). Nana Liu: Writing – original draft (equal); Writing – review & editing (equal).

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.