Quantum precision thermometry with weak measurement Arun Kumar Pati∗ and Chiranjib Mukhopadhyayy Quantum Information and Computation Group, Harish-Chandra Research Institute, HBNI, Allahabad 211019, India Sagnik Chakrabortyz and Sibasish Ghoshx Optics & Quantum Information Group, The Institute of Mathematical Sciences, HBNI, C. I. T. Campus, Taramani, Chennai - 600 113, India As the minituarization of electronic devices, which are sensitive to temperature, grows apace, sensing of temperature with ever smaller probes is more important than ever. Genuinely quantum mechanical schemes of thermometry are thus expected to be crucial to future technological progress. We propose a new method to measure the temperature of a bath using the weak measurement scheme with a finite dimensional probe. The precision offered by the present scheme not only shows similar qualitative features as the usual Quantum Fisher Information based thermometric protocols, but also allows for flexibility over setting the optimal thermometric window through judicious choice of post selection measurements. I. INTRODUCTION eral, complex numbers. In the last few years, weak values have been used in the context of measuring quantities which Quantum mechanics lends different status to observable may or may not have quantum mechanical observables as- quantities and parameters of a system, the latter of which are sociated with them, for example, the geometric phase [18], not expressible through Hermitian operators. Among phys- non Hermitian operators [19], density matrix corresponding ical parameters, temperature of a system is perhaps one of to a quantum state [20–24], or the entanglement content of a the most ubiquitously useful. Temperature drives chemical quantum state [25, 26]. The weak value amplification tech- reaction rates [1], control the generation of thermomelectric nique has found recent physical application in observations of current in thermocouples [2], or heat flow between different the spin Hall effect [27], photon trajectories [28], or the time baths [3]. Thus, thermometry, or the science of measuring delay between ultra fast processes [29] as well. Thus, it is temperature, is of paramount importance. This is especially reasonable to ask whether a weak measurement based scheme relevant for modern technological applications and devices, is a viable approach towards thermometry. In this paper, we where the thermal baths may themselves be relatively tiny. provide such a scheme. We mention here that there has been Nanoscale probes, which follow laws of quantum physics, are another theoretical as well as experimental paper by Egan and thus required to measure the temperature of such baths with- Stone [30][31] in the recent past, which introduces the con- out disturbing them too much. Quantum thermometry thus cept of weak thermometry. Experiments demonstrating weak aims at improving the technique of temperature sensing using value enhanced metrological tasks have also ben performed in small probes [4–10]. The analysis of quantum thermometry analogus contexts [32, 33]. so far has focused on seeking to find and saturate the quantum In the present work, we outline how to measure temper- Cramer Rao bound for various partially and fully thermalized ature using weak values with finite dimensional probes and configurations of systems [11, 12]. In addition to ensure non- specifically concentrate on qubit probes. We show the pres- invasiveness, these studies show possible improvement in the ence of an optimal temperature window for the precision of- precision due to quantum effects for non-equilbrium settings. fered by this scheme, which is a feature repeated in the usual Even in the steady state scenario, quantum enhancement in paradigm of quantum thermometry [11, 34–36]. One crucial the precision using a quantum switch has been recently re- advantage offered by the present scheme over previously con- ported [13]. For observables, another technique to experimen- sidered schemes is the ability to shift the optimal precision tally determine them quantitatively has recently come to the window while keeping the probe parameters fixed, and sim- fore, which relies on a so called weak measurement scheme ply changing the post selected state. In addition, we argue arXiv:1901.07415v2 [quant-ph] 16 Jun 2020 [14–17]. Weak measurement is a technique of ascertaining that, the proposal based on the weak value is suitable for mea- information about an observable through a weak interaction surement of a very hot body. If the temperature of a system between the system and the measurement apparatus gener- is very high, then prolonged contact with a measuring appa- ated by the observable. It is then followed by a strong post- ratus may damage the apparatus itself. On the other hand, in selection measurement on the system. One of the most well our weak measurement based scheme, the measuring appara- known facets of weak measurement scheme is the concept of tus is brought into contact with a thermalized qubit of the bath the so called weak values of an observable, which are, in gen- for a very short duration of time, thus potentially saving any damage to the apparatus. [37] The organization of the paper is as follows. SectionII ∗ [email protected] briefly reviews the weak measurement scheme. The main y [email protected] theme of our paper, namely the protocol for measuring tem- z [email protected] perature through weak values, is described in Section III. This x [email protected] is followed by the detailed analysis of precision offered by the 2 present scheme for a qubit probe in SectionIV. SectionV con- tains a complementary QFI based analysis of precision for the weak thermometric protocol. We finally conclude with a few observations and outline some possible future developments. T Apparatus Qubit II. WEAK MEASUREMENT In theory of weak measurements, a quantum system S is chosen to be in an initial state j iiS , called the pre-selected state, and it is made to interact with a measurement appara- FIG. 1. (color online) Schematic of the thermometric scheme. We tus M, prepared in a state jφiM. The interaction is weak in ˆ ˆ assume the bath being a collection of thermalized qubits, and couple strength, and generated by a Hamiltonian Hint = gA ⊗ Px, the measuring apparatus to one of the qubits, which is called the ˆ ˆ where A is a system observable, Px is the momentum operator probe qubit for a very short time. of M and g is a small positive number signifying the strength of the interaction. This is followed by a strong measurement on the system, called the post-selection measurement. The outcome of post-selection measurement that we focus on is degenerate energy eigenvalues E1; E2;:::; Ed and the corre- along the state j i. Clearly, the post-selection is a selective f sponding energy eigenstates j 1i ; j 2i ;:::; j di. Assume that measurement, as we ignore the other outcomes of the mea- S is in contact with a heat bath of temperature T and S has surement. −βHˆ −βHˆ reached the thermal equilibrium state ρT ≡ e = (Tr[e ]) = The time-evolved state of S + M before post-selection is Pd − Pd − e βEn j i h j = e βEn . Here β = 1=(k T), with given by n=1 n n n=1 B h i kB being taken as unity. Prepare the measuring aparatus M in −igAˆ⊗Pˆ x jΨi = e j ii ⊗ jφiM a state jφi, having position wave function φ(x). Note that here h i M is considered to be a continuous-variable system, in gen- ≈ ⊗ − ˆ ⊗ ˆ j i ⊗ j i IS IM igA Px i φ M ; (1) eral. We would like to perform measurement of an observable Aˆ on the system S . up to first order in g. Note that the state jΨ˜ iSM ≡ (IS ⊗ IM − ˆ ˆ igA ⊗ Px) j ii ⊗ jφiM is, in general, unnormalized. The re- Consider now evolution of S + M under the action of an duced state of jΨ˜ i on M is SM interaction Hamiltonian Hˆint = gAˆ ⊗ Pˆ x for a small amount of time τ. The interaction strength g is also considered to jφ˜ f i ≈ h f j ii − ig h f j Aˆ j i Pˆ x jφi ; (2) M i M be small – in the regime of weak interaction between S and which on normalizing looks like M. Here Pˆ x is the momentum observable of M canonically ˆ conjugate to the position observable Xˆ. We assume here that jφ i ≈ 1 − igA Pˆ jφi ≈ e−igAw Px jφi ; (3) f M w x M M during the time interval [0; τ], S and M are under the action only Hˆ where Aw ≡ h f j Aˆ j ii = h f j ii is called as the weak value of of the Hamiltonian int. This may be fulfilled in different ways: (i) We may decouple the system S at time t = 0 from Aˆ. Note that Aw can also be complex and take values beyond the eigenvalue range of the observable. the heat bath (after S achieves the thermal equilibrium state ρT ) and thereby switch on the interaction Hamiltonian Hˆint for Since the weak value Aw is, in general a complex quantity, one must experimentally observe the real as well as imagi- the time duration [0; τ]. (ii) On the other hand, we may think nary parts of the weak value. On measuring certain proper- of assuming here that the the strength g of interaction is much ˆ ties of jφ i , the real and imaginary parts of A can be deter- higher than that of the system Hamiltonian H, so that due to f M w ˆ mined. These properties include shift in position and momen- action of Hint for a small time span τ, it is enough to consider the change in states of S under the action of Hˆint only.